2007 AIME I Problem 3

Question

The complex number

z

is equal to

9 + bi,

where

b

is a positive real number and

i^2 = -1.

Given that the imaginary parts of

z^2

and

z^3

are equal, find

b.

Answer

Expand

(9 + bi)^2

and

(9 + bi)^3

with the binomial theorem and extract the imaginary parts

Both imaginary parts are divisible by

b;

cancel it before solving

By the binomial theorem,

z^2 = (81 - b^2) + 18bi

and

z^3 = (729 - 27b^2)

+ (243b - b^3)i.

Setting the imaginary parts equal gives

18b = 243b - b^3.

Since

b

is positive we may divide by

b,

leaving

b^2 = 243 - 18 = 225,

so

b = 15.

Problem 3 in other years